{
  "abstract": "We present a systematic development of the arithmetic of the class of linear orders under the ordered sum $(LO, +)$ and prove a number of new results. Our approach is based on a Euclidean algorithm for pairs of linear orders that almost additively commute. Among our results: (i.) We generalize and give unified proofs of the main classical theorems for $(LO, +)$, including Lindenbaum's division theorem for $(LO, +)$ and a representation theorem for additively commuting pairs of linear orders due to Aronszajn. (ii.) We solve the following problem, posed by Tarski in 1956: is it true that for every pair of linear orders $A, B$ and quadruple of natural numbers $n, m, k, l \\geq 1$, if $nA + mB \\cong kB + lA$ then $A + B \\cong B + A$? Tarski and Chang showed the answer is yes for certain choices of the coefficients $n, m, k, l$. We show the answer is yes in general. (iii.) We prove the following characterization of the additively commuting pairs in $LO$: $A + B \\cong B + A$ if and only if $ωA$ embeds initially in $ωB$ and $ω^* A$ embeds finally in $ω^* B$, or vice versa. We show this can be viewed as a correctly revised version of a refuted conjecture of Tarski. (iv.) We characterize the commutative semigroups $(S, \\oplus)$ that can be represented in $(LO, +)$ and show in particular they are all subsemigroups of naturally totally ordered semigroups in the sense of Clifford.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.20309",
  "author_total": 2,
  "authors": [
    {
      "name": "Garrett Ervin"
    },
    {
      "name": "Eric Paul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/76n8dcbe3759k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.20309",
  "fulltext_url": null,
  "id": "76n8dcbe3759k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.20309",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7203865628",
  "pdf_url": "https://arxiv.org/pdf/2608.20309",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/76n8dcbe3759k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2608.20309",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.20309"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The Additive Arithmetic of Linear Orders",
  "title_text": "The Additive Arithmetic of Linear Orders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/76n8dcbe3759k",
  "venue": "arXiv",
  "versions": []
}